Question: Simplify and expand the following expression: $ \dfrac{3}{n + 7}- \dfrac{1}{2n - 14}- \dfrac{2}{n^2 - 49} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the second term: $ \dfrac{1}{2n - 14} = \dfrac{1}{2(n - 7)}$ We can factor the quadratic in the third term: $ \dfrac{2}{n^2 - 49} = \dfrac{2}{(n + 7)(n - 7)}$ Now we have: $ \dfrac{3}{n + 7}- \dfrac{1}{2(n - 7)}- \dfrac{2}{(n + 7)(n - 7)} $ The least common multiple of the denominators is: $ (n + 7)(n - 7)$ In order to get the first term over $(n + 7)(n - 7)$ , multiply by $\dfrac{2(n - 7)}{2(n - 7)}$ $ \dfrac{3}{n + 7} \times \dfrac{2(n - 7)}{2(n - 7)} = \dfrac{6(n - 7)}{(n + 7)(n - 7)} $ In order to get the second term over $(n + 7)(n - 7)$ , multiply by $\dfrac{n + 7}{n + 7}$ $ \dfrac{1}{2(n - 7)} \times \dfrac{n + 7}{n + 7} = \dfrac{n + 7}{(n + 7)(n - 7)} $ In order to get the third term over $(n + 7)(n - 7)$ , multiply by $\dfrac{2}{2}$ $ \dfrac{2}{(n + 7)(n - 7)} \times \dfrac{2}{2} = \dfrac{4}{(n + 7)(n - 7)} $ Now we have: $ \dfrac{6(n - 7)}{(n + 7)(n - 7)} - \dfrac{n + 7}{(n + 7)(n - 7)} - \dfrac{4}{(n + 7)(n - 7)} $ $ = \dfrac{ 6(n - 7) - (n + 7) - 4} {(n + 7)(n - 7)} $ Expand: $ = \dfrac{6n - 42 - n - 7 - 4}{2n^2 - 98} $ $ = \dfrac{5n - 53}{2n^2 - 98}$